You should look at this ncatlab exposition on "convenient categories of topological spaces", and references. One problem is still the difficulty of getting a locally cartesian closed convenient category; Spanier's quasicategories seemed a possibility but have not been taken up, partly because the quasicategories on the 2 point set formed a class (comment from E. Dyer).

You should look at this ncatlab exposition on "convenient categories of topological spaces", and references. One problem is still the difficulty of getting a locally cartesian closed convenient category; Spanier's "quasitopologies" seemed a possibility but have not been taken up, partly because the quasicategories on the 2 point set formed a class (comment from E. Dyer).